#### Transcript Section 6.4 – Logarithmic Functions

```Section 6.3 โ Exponential Functions
Laws of Exponents
If s, t, a, and b are real numbers where a > 0 and b > 0, then:
๐ ๐  โ ๐๐ก = ๐ ๐ +๐ก
๐ก
1 =1
(๐ ๐  )๐ก = ๐ ๐ ๐ก
(๐๐)๐ก = ๐๐ก ๐ ๐ก
0
๐ =1
๐โ๐ก
1
1
= ๐ก=
๐
๐
Definition:
An Exponential Function is in the form, ๐ ๐ฅ = ๐ถ๐ ๐ฅ
โaโ is a positive real number and does not equal 1
โaโ is the base and is the Growth Factor
โCโ is a real number and does not equal 0
โCโ is the Initial Value because ๐ 0 = ๐ถ๐0 = ๐ถ
The domain of f(x) is the set of all real numbers
๐(๐ฅ + 1)
๐ถ๐ ๐ฅ+1 ๐ถ๐ ๐ฅ ๐1
=๐
โ
=
=๐
๐ฅ
๐ฅ
๐(๐ฅ)
๐ถ๐
๐ถ๐
๐ก
Section 6.3 โ Exponential Functions
Examples
๐ฅ
๐(๐ฅ)
๐(๐ฅ + 1)
=๐
๐(๐ฅ)
โ1
2
3
1
3
=
2/3 2
0
1
3/2 3
=
1
2
1
3
2
9/4 3
=
3/2 2
2
9
4
27/8 3
=
9/4
2
3
27
8
๐=
๐ ๐ฅ = ๐ถ๐ ๐ฅ
๐ 0 = 1, ๐กโ๐๐๐๐๐๐๐ ๐ถ = 1
3
๐ ๐ฅ =1
2
๐ฅ
3
2
3
=
2
๐ฅ
Section 6.3 โ Exponential Functions
Examples
๐ฅ
๐(๐ฅ)
๐(๐ฅ + 1)
=๐
๐(๐ฅ)
โ1
1
2
1/4 1
=
1/2 2
0
1
4
1/8 1
=
1/4 2
1
1
8
1/16 1
=
1/8
2
2
1
16
1/32 1
=
1/16 2
3
1
32
๐=
๐ 0 = 1/4, ๐กโ๐๐๐๐๐๐๐ ๐ถ = 1/4
1 3
๐ ๐ฅ =
4 2
๐ฅ
1
2
๐ ๐ฅ = ๐ถ๐ ๐ฅ
Section 6.3 โ Exponential Functions
Properties of the Exponential Function,๐ ๐ฅ = ๐ ๐ฅ , ๐ > 1
The domain is the set of all real numbers.
The range is the set of all positive real numbers.
The y-intercept is 1; x-intercepts do not exist.
The x-axis (y = 0) is a horizontal asymptote, as x๏ฎ๏ญ๏ฅ.
If a > 1, the f(x) is increasing function.
The graph contains the points (0, 1), (1, a), and (-1, 1/a).
The graph is smooth and continuous.
Section 6.3 โ Exponential Functions
๐ ๐ฅ = ๐๐ฅ , ๐ > 1
The graph of the exponential function is shown below.
๐ฆ = ๐ ๐ฅ = ๐๐ฅ
a
Section 6.3 โ Exponential Functions
Properties of the Exponential Function, ๐ ๐ฅ = ๐ ๐ฅ , 0 < ๐ < 1
The domain is the set of all real numbers.
The range is the set of all positive real numbers.
The y-intercept is 1; x-intercepts do not exist.
The x-axis (y = 0) is a horizontal asymptote, as x๏ฎ๏ฅ.
If 0 < a < 1, then f(x) is a decreasing function.
The graph contains the points (0, 1), (1, a), and (-1, 1/a).
The graph is smooth and continuous.
Section 6.3 โ Exponential Functions
๐ ๐ฅ = ๐๐ฅ , 0 < ๐ < 1
The graph of the exponential function is shown below.
๐ ๐ฅ = ๐๐ฅ , 0 < ๐ < 1
Section 6.3 โ Exponential Functions
Eulerโs Constant โ e
The value of the following expression approaches e,
1
1+
๐
๐
as n approaches ๏ฅ.
Using calculus notation,
1
๐ = lim 1 +
๐โโ
๐
๐
Applications of e
Growth and decay
Compound interest
Differential and Integral calculus with exponential functions
Infinite series
Section 6.3 โ Exponential Functions
Theorem
If ๐๐ข = ๐๐ฃ , then ๐ข = ๐ฃ.
Solving Exponential Equations
1) 32๐ฅโ1 = 3๐ฅ
2๐ฅ โ 1 = ๐ฅ
x=1
2) 22๐ฅ = 32
22๐ฅ = 4 โ 8
3) 4๐ฅ+2 = 8
22
๐ฅ+2
= 23
22๐ฅ = 22 โ 23
22(๐ฅ+2) = 23
22๐ฅ = 25
2๐ฅ = 5
5
๐ฅ=
2
22๐ฅ+4 = 23
2๐ฅ + 4 = 3
1
๐ฅ=โ
2
Section 6.3 โ Exponential Functions
Solving Exponential Equations
4)
1 ๐ฅ+2
3
3โ1
93๐ฅ
=
๐ฅ+2
= 32
3โ๐ฅโ2 = 36๐ฅ
โ๐ฅ โ 2 = 6๐ฅ
๐ฅ=โ
2
7
3๐ฅ
5) 81 โ 9โ2๐โ2 = 27
9 โ 9 โ 9โ2๐โ2 = 27
32 โ 32 32 โ2๐โ2 = 33
34 โ 3โ4๐ฅโ4 = 33
3โ4๐ฅ = 33
โ4๐ฅ = 3
3
๐ฅ=โ
4
Section 6.4 โ Logarithmic Functions
The exponential and logarithmic functions are inverses of each other.
The logarithmic function is defined by
๐ฆ = ๐๐๐๐ ๐ฅ
๐๐ ๐๐๐ ๐๐๐๐ฆ ๐๐
๐ฅ = ๐๐ฆ
The domain is the set of all positive real numbers 0, โ .
The range is the set of all real numbers โโ, โ .
The x-intercept is 1 and the y-intercept does not exist.
The y-axis (x = 0) is a vertical asymptote.
If 0 < a < 1, then the logarithmic function is a decreasing function.
If a > 1, then the logarithmic function is an increasing function.
The graph contains the points (1, 0), (a, 1), and (1/a, โ1).
The graph is smooth and continuous.
Section 6.4 โ Logarithmic Functions
The graph of the logarithmic function is shown below.
๐, 1
๐ฆ = ๐๐๐๐ ๐ฅ
๐
The natural logarithmic function
๐ฆ = ๐๐๐๐ ๐ฅ = ln ๐ฅ
๐๐ ๐๐๐ ๐๐๐๐ฆ ๐๐
๐ฅ = ๐๐ฆ
๐๐ ๐๐๐ ๐๐๐๐ฆ ๐๐
๐ฅ = 10๐ฆ
The common logarithmic function
๐ฆ = ๐๐๐10 ๐ฅ = ๐๐๐ ๐ฅ
Section 6.4 โ Logarithmic Functions
Graphs of ๐ ๐ฅ = ๐๐๐๐ ๐ฅ ๐๐๐ ๐ ๐ฅ = ๐ ๐ฅ
๐, 1
๐ฆ = ๐๐๐๐ ๐ฅ
๐
๐ฆ = ๐ ๐ฅ = ๐๐ฅ
a
Section 6.4 โ Logarithmic Functions
Graphs of ๐ ๐ฅ = ๐๐๐๐ ๐ฅ ๐๐๐ ๐ ๐ฅ = ๐ ๐ฅ
Inverse Functions: ๐ ๐ฅ = ๐๐๐๐ ๐ฅ ๐๐๐ ๐ ๐ฅ = ๐ ๐ฅ
๐ฆ = ๐ ๐ฅ = ๐๐ฅ
a
๐, 1
๐
๐ฆ = ๐๐๐๐ ๐ฅ
Section 6.4 โ Logarithmic Functions
Graphs of ๐ ๐ฅ = ๐ ๐ฅ ๐๐๐ ๐ ๐ฅ = ln ๐ฅ
Inverse Functions: ๐ ๐ฅ = ๐ ๐ฅ ๐๐๐ ๐ ๐ฅ = ln ๐ฅ
Section 6.4 โ Logarithmic Functions
Graph ๐ ๐ฅ = ๐ ๐ฅ + 2
๐ฆ = ๐ ๐ฅ = ๐๐ฅ
a
๐, 1
๐
๐ฆ = ๐๐๐๐ ๐ฅ
Section 6.4 โ Logarithmic Functions
Graph ๐ ๐ฅ = ๐ ๐ฅ+1 + 2
๐ฆ = ๐ ๐ฅ = ๐๐ฅ
a
๐, 1
๐
๐ฆ = ๐๐๐๐ ๐ฅ
Section 6.4 โ Logarithmic Functions
Graph ๐ ๐ฅ = ๐๐๐๐ ๐ฅ โ 1
๐ฆ = ๐ ๐ฅ = ๐๐ฅ
a
๐, 1
๏ท
๐
๐ฆ = ๐๐๐๐ ๐ฅ
Section 6.4 โ Logarithmic Functions
Graph ๐ ๐ฅ = ๐๐๐๐ (๐ฅ โ 2) โ 1
๐ฆ = ๐ ๐ฅ = ๐๐ฅ
a
๐, 1
๐
๐ฆ = ๐๐๐๐ ๐ฅ
๏ท
Section 6.4 โ Logarithmic Functions
Change the exponential statements to logarithmic statements
๐5 = 6.7
8๐ฅ = 9.2
๐3 = ๐
๐๐ฅ = 4
5 = ๐๐๐๐ 6.7
๐ฅ = ๐๐๐8 6.7
3 = ln ๐
๐ฅ = ln 4
Change the logarithmic statements to exponential statements
๐๐๐3 5 = ๐ฅ
๐๐๐๐ฅ 7 = 4
ln ๐ = 6
5 = 3๐ฅ
7 = ๐ฅ4
๐ = ๐6
Solve the following equations
๐๐๐3 (2๐ฅ) = 1
๐๐๐2 (5๐ฅ + 1) = 4
๐๐๐2 32 = โ3๐ฅ + 9
2๐ฅ = 31
3
๐ฅ=
2
5๐ฅ + 1 = 24
32 = 2โ3๐ฅ+9
5๐ฅ + 1 = 16
25 = 2โ3๐ฅ+9
5๐ฅ = 15
๐ฅ=3
5 = โ3๐ฅ + 9
โ4 = โ3๐ฅ
4
=๐ฅ
3
Section 6.4 โ Logarithmic Functions
Solve the following equations
๐ ๐ฅ = 10
๐ 7๐ฅ = 15
8 + 2๐ ๐ฅ = 12
๐ฅ = ln 10
7๐ฅ = ln 15
ln 15
๐ฅ=
7
2๐ ๐ฅ = 4
๐๐ฅ = 2
๐ฅ = ln 2
```